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Mirrors > Home > MPE Home > Th. List > ressmplvsca | Structured version Visualization version GIF version |
Description: A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.) |
Ref | Expression |
---|---|
ressmpl.s | β’ π = (πΌ mPoly π ) |
ressmpl.h | β’ π» = (π βΎs π) |
ressmpl.u | β’ π = (πΌ mPoly π») |
ressmpl.b | β’ π΅ = (Baseβπ) |
ressmpl.1 | β’ (π β πΌ β π) |
ressmpl.2 | β’ (π β π β (SubRingβπ )) |
ressmpl.p | β’ π = (π βΎs π΅) |
Ref | Expression |
---|---|
ressmplvsca | β’ ((π β§ (π β π β§ π β π΅)) β (π( Β·π βπ)π) = (π( Β·π βπ)π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressmpl.u | . . . . 5 β’ π = (πΌ mPoly π») | |
2 | eqid 2733 | . . . . 5 β’ (πΌ mPwSer π») = (πΌ mPwSer π») | |
3 | ressmpl.b | . . . . 5 β’ π΅ = (Baseβπ) | |
4 | eqid 2733 | . . . . 5 β’ (Baseβ(πΌ mPwSer π»)) = (Baseβ(πΌ mPwSer π»)) | |
5 | 1, 2, 3, 4 | mplbasss 21426 | . . . 4 β’ π΅ β (Baseβ(πΌ mPwSer π»)) |
6 | 5 | sseli 3944 | . . 3 β’ (π β π΅ β π β (Baseβ(πΌ mPwSer π»))) |
7 | eqid 2733 | . . . 4 β’ (πΌ mPwSer π ) = (πΌ mPwSer π ) | |
8 | ressmpl.h | . . . 4 β’ π» = (π βΎs π) | |
9 | eqid 2733 | . . . 4 β’ ((πΌ mPwSer π ) βΎs (Baseβ(πΌ mPwSer π»))) = ((πΌ mPwSer π ) βΎs (Baseβ(πΌ mPwSer π»))) | |
10 | ressmpl.2 | . . . 4 β’ (π β π β (SubRingβπ )) | |
11 | 7, 8, 2, 4, 9, 10 | resspsrvsca 21410 | . . 3 β’ ((π β§ (π β π β§ π β (Baseβ(πΌ mPwSer π»)))) β (π( Β·π β(πΌ mPwSer π»))π) = (π( Β·π β((πΌ mPwSer π ) βΎs (Baseβ(πΌ mPwSer π»))))π)) |
12 | 6, 11 | sylanr2 682 | . 2 β’ ((π β§ (π β π β§ π β π΅)) β (π( Β·π β(πΌ mPwSer π»))π) = (π( Β·π β((πΌ mPwSer π ) βΎs (Baseβ(πΌ mPwSer π»))))π)) |
13 | 3 | fvexi 6860 | . . . 4 β’ π΅ β V |
14 | 1, 2, 3 | mplval2 21425 | . . . . 5 β’ π = ((πΌ mPwSer π») βΎs π΅) |
15 | eqid 2733 | . . . . 5 β’ ( Β·π β(πΌ mPwSer π»)) = ( Β·π β(πΌ mPwSer π»)) | |
16 | 14, 15 | ressvsca 17233 | . . . 4 β’ (π΅ β V β ( Β·π β(πΌ mPwSer π»)) = ( Β·π βπ)) |
17 | 13, 16 | ax-mp 5 | . . 3 β’ ( Β·π β(πΌ mPwSer π»)) = ( Β·π βπ) |
18 | 17 | oveqi 7374 | . 2 β’ (π( Β·π β(πΌ mPwSer π»))π) = (π( Β·π βπ)π) |
19 | fvex 6859 | . . . . 5 β’ (Baseβπ) β V | |
20 | ressmpl.s | . . . . . . 7 β’ π = (πΌ mPoly π ) | |
21 | eqid 2733 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
22 | 20, 7, 21 | mplval2 21425 | . . . . . 6 β’ π = ((πΌ mPwSer π ) βΎs (Baseβπ)) |
23 | eqid 2733 | . . . . . 6 β’ ( Β·π β(πΌ mPwSer π )) = ( Β·π β(πΌ mPwSer π )) | |
24 | 22, 23 | ressvsca 17233 | . . . . 5 β’ ((Baseβπ) β V β ( Β·π β(πΌ mPwSer π )) = ( Β·π βπ)) |
25 | 19, 24 | ax-mp 5 | . . . 4 β’ ( Β·π β(πΌ mPwSer π )) = ( Β·π βπ) |
26 | fvex 6859 | . . . . 5 β’ (Baseβ(πΌ mPwSer π»)) β V | |
27 | 9, 23 | ressvsca 17233 | . . . . 5 β’ ((Baseβ(πΌ mPwSer π»)) β V β ( Β·π β(πΌ mPwSer π )) = ( Β·π β((πΌ mPwSer π ) βΎs (Baseβ(πΌ mPwSer π»))))) |
28 | 26, 27 | ax-mp 5 | . . . 4 β’ ( Β·π β(πΌ mPwSer π )) = ( Β·π β((πΌ mPwSer π ) βΎs (Baseβ(πΌ mPwSer π»)))) |
29 | ressmpl.p | . . . . . 6 β’ π = (π βΎs π΅) | |
30 | eqid 2733 | . . . . . 6 β’ ( Β·π βπ) = ( Β·π βπ) | |
31 | 29, 30 | ressvsca 17233 | . . . . 5 β’ (π΅ β V β ( Β·π βπ) = ( Β·π βπ)) |
32 | 13, 31 | ax-mp 5 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) |
33 | 25, 28, 32 | 3eqtr3i 2769 | . . 3 β’ ( Β·π β((πΌ mPwSer π ) βΎs (Baseβ(πΌ mPwSer π»)))) = ( Β·π βπ) |
34 | 33 | oveqi 7374 | . 2 β’ (π( Β·π β((πΌ mPwSer π ) βΎs (Baseβ(πΌ mPwSer π»))))π) = (π( Β·π βπ)π) |
35 | 12, 18, 34 | 3eqtr3g 2796 | 1 β’ ((π β§ (π β π β§ π β π΅)) β (π( Β·π βπ)π) = (π( Β·π βπ)π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3447 βcfv 6500 (class class class)co 7361 Basecbs 17091 βΎs cress 17120 Β·π cvsca 17145 SubRingcsubrg 20260 mPwSer cmps 21329 mPoly cmpl 21331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-tset 17160 df-subg 18933 df-ring 19974 df-subrg 20262 df-psr 21334 df-mpl 21336 |
This theorem is referenced by: ressply1vsca 21626 |
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